I do not believe that Paul McCartney was killed in a car accident in 1966, and replaced with a look-alike.
According to this conspiracy theory, the man currently known as Paul McCartney isn't really Paul McCartney. This is because the original Paul McCartney, who played in a group called the Quarrymen, which later evolved into the Beatles, died in a car accident in late 1966. The Beatles found a replacement in William Shears Campbell, a Scottish man who (depending on which version of the story you run across) had either won or came in second in a Paul McCartney look-alike contest. Starting in 1967, the band started passing off Campbell as McCartney, perhaps with the help of some cosmetic surgery. They also started to incorporate, into their songs and album cover artwork, clues to Paul's alleged death and replacement. One early example of this would be the lyrics "the one and only Billy Shears" in the title song of their 1967 album Sargeant Pepper's Lonely Hearts Club Band, allegedly giving a clue to Campbell's identity. If this theory is true, it would mean that a man named Bill Campbell, after assuming McCartney's identity, married Linda Eastman and later Heather Mills, founded the band Wings, had a long solo career, was knighted, and is still around today. Meanwhile, the "true" Paul McCartney has been pushing up daisies for fifty years (although no believer in all this, as far as I can tell, has ever tried to point out the location of his grave).
The reason I do not believe that today's Paul McCartney is really William Shears Campbell is in the details of what the two men would have to have had in common, in order for one to truly replace the other. First, Campbell would have to physically resemble McCartney, not just in the face, but in his overall build. The two men would have to be close in height, weight and age. They would also have to have similar singing voices, both in range and tonal quality. At best, Campbell would have be able to imitate McCartney's singing and speech, in a manner analogous to Elvis impersonators. Campbell would also have to be able to play the same musical instruments as McCartney, including left-handed guitar and bass. (Let's not forget that left-handedness is relatively rare.) Finally, Campbell would have to be able to compose songs in the same general style as McCartney had, with the help of John Lennon. This would mean that Penny Lane, Hey Jude, Lady Madonna and Let It Be, as well as post-Beatles songs such as Uncle Albert/Admiral Halsey, were written or co-written by Campbell instead of McCartney. I can see that someone possibly could have one or maybe two of these attributes in common with McCartney, but the more things they have in common, the more unlikely the coincidence becomes. For a relatively unknown person to have all this in common with McCartney, without being his long-lost identical twin, would be, in my opinion, just about impossible.
One type of alleged evidence used in support of this theory consists of photographs showing McCartney playing right-handed guitar. As the argument goes, Campbell was right-handed, so these pictures are of Campbell, not McCartney. Such photos, however, can be produced by taking one of a left-handed guitarist, and flipping it horizontally. Moreover, during his numerous live performances after the time of his "death", McCartney has always been shown playing left-handed, together with his right-handed band-mates, such as John Lennon, George Harrison, and Denny Laine.
I'm also unaware of any independent evidence for Campbell's pre-1967 existence, let alone musical talents. If Campbell had the ability to compose songs as good as those mentioned above, where was he before 1967? Was he simply a talented but undiscovered musician? On the other hand, I can believe that the Beatles might have wanted people to think that Paul may have died and been replaced with someone else, as some sort of mind game. If so, it may be hard to determine where they actually put clues into their songs or cover art, and where fans have read clues into them.
Another component of this conspiracy theory is that the police and media were all induced, perhaps by bribery, to cover up McCartney's fatal accident. This would mean that in order to believe in the McCartney/Campbell switch, you'd have to believe that the cover-up of the accident has been successful for fifty years, without anyone snitching, starting with the police officers who arrived on the scene. Policeman, after all, have the job of investigating and documenting such accidents, not covering them up. It's also hard to believe that if the media received credible information that a celebrity of McCartney's magnitude had died in a car wreck, that somehow everyone in the media would refrain from publishing the story. Besides all that, there doesn't seem to be any reports of grief from the other Beatles. If you're in the most successful band in the world, and you and your band-mates have all known each other since your teenage years, and then one of you suddenly dies, there is no way that you will not be greatly affected emotionally.
There is one way to put this theory to rest, once and for all, which is a Y-chromosome DNA test between Paul McCartney and his brother Michael, known as Mike McGear. The McCartney brothers, if they really are brothers, would have the same Y-chromosome, inherited from their father. Campbell's Y-chromosome, if today's Paul really is Campbell, would be different. But as far as I'm concerned, such a test would most likely confirm all that I've said above.